The function is defined by . What is the -coordinate of the positive -intercept of the graph of ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 130 | Free SAT Prep | Aniko

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Question 130·Easy·Nonlinear Functions

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The function $f$ is defined by $f(x) = \sqrt{9 - x^2}$ .
What is the $x$ -coordinate of the positive $x$ -intercept of the graph of $f$ ?

Your answer

(Express the answer as an integer)

For x-intercept questions on the SAT, always remember that the y-value is 0 at an intercept, so set the function equal to 0 and solve for x. With square root functions like $f(x) = \sqrt{9 - x^2}$ , set the square root equal to 0, then set the inside (the radicand) equal to 0 and solve. Finally, check whether the problem wants a specific intercept (positive, negative, or all) and choose the solution(s) that match that condition.

Hints

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Connect x-intercept to the function value

At an x-intercept, what is the y-value of the point? How does that relate to $f(x)$ ?

Write an equation for the intercept

Set $f(x) = \sqrt{9 - x^2}$ equal to 0 and think about when a square root expression can be 0.

Solve and then interpret the solutions

Solving $9 - x^2 = 0$ will give you two x-values. After you find them, look back at the question to decide which one you actually need.

Desmos Guide

Graph the function

In Desmos, type y = sqrt(9 - x^2) to graph the function $f(x)$ .

Show the x-axis as a reference

Type y = 0 to draw the x-axis as a horizontal line (this is already the x-axis, but having it as an equation makes intersections easy to see).

Identify the positive x-intercept

Look for the point where the graph of $y = \sqrt{9 - x^2}$ meets the x-axis on the right side of the origin. Click that intersection point and read off the x-coordinate shown by Desmos; that is the positive x-intercept.

Step-by-step Explanation

Use the definition of an x-intercept

An x-intercept is a point where the graph crosses the x-axis.

On the x-axis, the y-value is 0. Here, the y-value is $f(x)$ , so at an x-intercept we must have

f(x) = 0.

Since $f(x) = \sqrt{9 - x^2}$ , set this equal to 0.

Set the function equal to 0 and remove the square root

Write the equation for an x-intercept:

\sqrt{9 - x^2} = 0.

A square root is 0 only when the expression inside it is 0. That means

9 - x^2 = 0.

Solve the equation for x

Now solve $9 - x^2 = 0$ :

\begin{aligned} 9 - x^2 &= 0 \\ -x^2 &= -9 \\ x^2 &= 9. \end{aligned}

Take the square root of both sides to find the possible x-values.

Choose the positive x-intercept

From $x^2 = 9$ , the solutions are

x = 3 \quad \text{or} \quad x = -3.

The question asks for the positive x-intercept, so we choose $x = 3$ . The x-coordinate of the positive x-intercept is 3.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation